I wonder how to formally exact write this expansion. To understand this, I would like to know

What is the independent variable in Eq. 17.46? Is it $\phi$, $\mu$ or $(e\phi + \mu)$?

Does it make sense to write it as
$$\rho^{ind}(\textbf{r}) = -e \left[ n_0(\mu) + \left . e\phi(\textbf{r})\frac{\partial n_0(e\phi+\mu)}{\partial \mu}\right|_{e\phi=0} - n_0(\mu) \right]$$
But then, if I set $e\phi = 0$, would the middle term not collapse to zero?

Let's answer both questions at once. You're interested in the behaviour of the function $n_0(\mu + e\phi)$ as you vary $\phi$, especially in the limit $e \phi \rightarrow 0.$ This suggests that you expand $n_0(\mu + e\phi)$ around $e\phi = 0$: